Elementary Set Theory Proof/Identity Relation

Prove: R is a reflexive relation on A if and only if the identity relation on A is a subset of R.

I know that the proof should be in form:

(i) Showing that if R is a reflexive relation on A, then the identity relation on A is a subset of R.

(ii) If the identity relation is a subset of R, then R is a reflexive relation on A.

I am having trouble constructing the second part properly to show that for all x in A, xRx.

Solution 1:

Let $R$ be a relation on $A$, ie $R\subset A\times A$.

The identity relation on $A$ is $I = \{(x,x)~|~x\in A\}$.

• "$R$ is reflexive" means exactly $\forall x\in A,~(x,x)\in R$.
• "$I \subset R$ means exactly $\forall (x,y)\in I,~(x,y)\in R$. But everyone in $I$ is a $(x,x)$, so $I\subset R$ means $\forall x\in A,~(x,x)\in R$.